Hmm. I think I may have figured out what's going on. It seems like Faulhaber's formula fails to sum

for what appears to be . Below that it works and sums it to, apparently . So that would provide an analytic continuation to higher q-values, thus the formula given is valid in a sense, but direct application of the Faulhaber formula will not work. It seems that a combination of analytic continuation and divergent summation theory applied to Faulhaber and transseries theory is needed to put together a comprehensive theory of continuum sums of general analytic functions.

I find the appearance of interesting, considering it is the magnitude of the imaginary period of the exponential function. I don't know if that is significant or not.